Mathematical Induction!!! (1 Viewer)

[plshelp]

hey i need help again...

The question is
Prove this divisibility result, for even n
n3 (cubed) + 2n is divisible by 12

 

[mirakon]

1. Prove it for n=1. Im sure u can do this yourself
2. Assume it is true for n=k. Again u can state this i am sure. You should have something like:

k^3+2k=12m where m is some integer

3. Prove it true for n=k+2. This is because n is even (as in the question) and assuming for n=k+1 would mean n is odd. I'm assuming this is where you're having problems? Ok then. You''re aiming to prove

(k+2)^3+2(k+2)= 12xsomething

LHS= (k+2)^3+2(k+2)

= k^3 + 6k^2 +12k +8 +2k +4

=12m + 6k^2+ 12k + 12

now 6=3x2 therefore 6k^2=3(2xk^2) as k^2 is even therefore 2xeven= something divsible by 4 (this should be clear). As such 2k^2=4j where j is random integer

Therefore

=12m +12j +12k +12

=12(m+j+k+1)
= RHS

4. Just state the standard proof statement.

 

[random-1006]

1. Prove it for n=1. Im sure u can do this yourself
2. Assume it is true for n=k. Again u can state this i am sure. You should have something like:

k^3+2k=12m where m is some integer

3. Prove it true for n=k+2. This is because n is even (as in the question) and assuming for n=k+1 would mean n is odd. I'm assuming this is where you're having problems? Ok then. You''re aiming to prove

(k+2)^3+2(k+2)= 12xsomething

LHS= (k+2)^3+2(k+2)

= k^3 + 6k^2 +12k +8 +2k +4

=12m + 6k^2+ 12k + 12

now 6=3x2 therefore 6k^2=3(2xk^2) as k^2 is even therefore 2xeven= something divsible by 4 (this should be clear). As such 2k^2=4j where j is random integer

Therefore

=12m +12j +12k +12

=12(m+j+k+1)
= RHS

4. Just state the standard proof statement.

huh havent seen one like this before, i kept thinking it was a typo lol

 

[x jiim]

mirakon, wouldn't you start with n=2, since it's for even n?

 

[random-1006]

mirakon, wouldn't you start with n=2, since it's for even n?

yes you would

 

[mirakon]

yes you would

whoops lol
skipped that step and went straight for the 'real' induction, thought OP could do this herself. thanks for the correction

 

[plshelp]

thanx guys